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Computer Science > Learning

Title:
Uncovering the Riffled Independence Structure of Rankings

Abstract: Representing distributions over permutations can be a daunting task due to
the fact that the number of permutations of $n$ objects scales factorially in
$n$. One recent way that has been used to reduce storage complexity has been to
exploit probabilistic independence, but as we argue, full independence
assumptions impose strong sparsity constraints on distributions and are
unsuitable for modeling rankings. We identify a novel class of independence
structures, called \emph{riffled independence}, encompassing a more expressive
family of distributions while retaining many of the properties necessary for
performing efficient inference and reducing sample complexity. In riffled
independence, one draws two permutations independently, then performs the
\emph{riffle shuffle}, common in card games, to combine the two permutations to
form a single permutation. Within the context of ranking, riffled independence
corresponds to ranking disjoint sets of objects independently, then
interleaving those rankings. In this paper, we provide a formal introduction to
riffled independence and present algorithms for using riffled independence
within Fourier-theoretic frameworks which have been explored by a number of
recent papers. Additionally, we propose an automated method for discovering
sets of items which are riffle independent from a training set of rankings. We
show that our clustering-like algorithms can be used to discover meaningful
latent coalitions from real preference ranking datasets and to learn the
structure of hierarchically decomposable models based on riffled independence.